## The Moment of Inertia of a Body About an Axis

The moment of Inertia of a body about a point, labelledis found by summing or integrating the moments of inertia of the particles that constitute the body:

for a collection of discrete particles with massesat positions

wherewithequal to the mass of the small regionof densitywhereis the mass of the body at the point and the integral is carried out over the space(which may be a line, area or volume) occupied by the body

For example, the moment of inertia of a circular piece of card of uniform mass per unit area and radiusabout an axis perpendicular to the card passing through the centre can be found by integration.

From the diagram above,sohence

The mass of the card isso